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  4. If p and q are positive real numbers such that p2 + q2 = 1, then the maximum value of (p + q) is

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Q. If $p$ and $q$ are positive real numbers such that $p^2 + q^2 = 1,$ then the maximum value of $(p + q)$ is

AIEEEAIEEE 2007Trigonometric Functions

Solution:

Using $A.M. \ge G.M.$
$\frac{p^{2}+q^{2}}{2}\ge pq$
$\Rightarrow pq \le \frac{1}{2}$
$\left(p+q\right)^{2}=p^{2}+q^{2}+2pq$
$\left(p+q\right)^{2}=p^{2}+q^{2}+2pq$
$\Rightarrow p+q \le \sqrt{2}.$